Schubert calculus of Richardson varieties stable under spherical Levi subgroups

We observe that the expansion in the basis of Schubert cycles for $H^*(G/B)$ of the class of a Richardson variety stable under a spherical Levi subgroup is described by a theorem of Brion. Using this observation, along with a combinatorial model of the poset of certain symmetric subgroup orbit closures, we give positive combinatorial descriptions of certain Schubert structure constants on the full flag variety in type $A$. Namely, we describe $c_{u,v}^w$ when $u$ and $v$ are inverse to Grassmannian permutations with unique descents at $p$ and $q$, respectively. We offer some conjectures for similar rules in types $B$ and $D$, associated to Richardson varieties stable under spherical Levi subgroups of SO(2n+1,\C) and SO(2n,\C), respectively.Comment: Section 4 significantly shortened, and other minor changes made as suggested by referees. Final version, to appear in Journal of Algebraic Combinatoric

Lectures on conformal field theory and Kac-Moody algebras

This is an introduction to the basic ideas and to a few further selected topics in conformal quantum field theory and in the theory of Kac-Moody algebras.Comment: 59 pages, LaTeX2e, extended version of lectures given at the Graduate Course on Conformal Field Theory and Integrable Models (Budapest, August 1996), to appear in Springer Lecture Notes in Physic

The Physarum polycephalum Genome Reveals Extensive Use of Prokaryotic Two-Component and Metazoan-Type Tyrosine Kinase Signaling

Physarum polycephalum is a well-studied microbial eukaryote with unique experimental attributes relative to other experimental model organisms. It has a sophisticated life cycle with several distinct stages including amoebal, flagellated, and plasmodial cells. It is unusual in switching between open and closed mitosis according to specific life-cycle stages. Here we present the analysis of the genome of this enigmatic and important model organism and compare it with closely related species. The genome is littered with simple and complex repeats and the coding regions are frequently interrupted by introns with a mean size of 100 bases. Complemented with extensive transcriptome data, we define approximately 31,000 gene loci, providing unexpected insights into earlyeukaryoteevolution.Wedescribeextensiveuseofhistidinekinase-basedtwo-componentsystemsandtyrosinekinasesignaling, the presence of bacterial and plant type photoreceptors (phytochromes, cryptochrome, and phototropin) and of plant-type pentatricopeptide repeat proteins, as well as metabolic pathways, and a cell cycle control system typically found in more complex eukaryotes. Our analysis characterizes P. polycephalum as a prototypical eukaryote with features attributed to the last common ancestor of Amorphea, that is, the Amoebozoa and Opisthokonts. Specifically, the presence of tyrosine kinases inAcanthamoeba and Physarum as representatives of two distantly related subdivisions ofAmoebozoa argues against the later emergence of tyrosine kinase signaling in the opisthokont lineage and also against the acquisition by horizontal gene transfe

Identification of a Rare Coding Variant in Complement 3 Associated with Age-related Macular Degeneration

Macular degeneration is a common cause of blindness in the elderly. To identify rare coding variants associated with a large increase in risk of age-related macular degeneration (AMD), we sequenced 2,335 cases and 789 controls in 10 candidate loci (57 genes). To increase power, we augmented our control set with ancestry-matched exome sequenced controls. An analysis of coding variation in 2,268 AMD cases and 2,268 ancestry matched controls revealed two large-effect rare variants; previously described R1210C in the CFH gene (fcase = 0.51%, fcontrol = 0.02%, OR = 23.11), and newly identified K155Q in the C3 gene (fcase = 1.06%, fcontrol = 0.39%, OR = 2.68). The variants suggest decreased inhibition of C3 by Factor H, resulting in increased activation of the alternative complement pathway, as a key component of disease biology